Investigating the mechanisms behind the Seebeck-Peltier effect of electrothermophoresis, Onsager found his famous “reciprocal relations of irreversible processes” [1]}. Following his notion, a quantity called entropy production can be defined which is responsible for the relaxation of the system to equilibrium. For any thermodynamic observable \(X\)  it is proportional to the temporal derivative of \(X\)  times the thermodynamic force of \(X\) , which is the derivative of the entropy with respect to \(X\) . For continuous systems, one can introduce the dependency of both on space [2]}. Onsager's ansatz is then that the spatial flux of \(X\)  is linearly dependent on the force itself with a proportionality factor \(L>0\) , introducing his linear phenomenological equations. In our case, we are interested in the fluxes of heat and particles. The driving force for the dissipation of heat can be found to be \(\nabla (T^{-1})\)  [2]}, while that of the motion of particles can be found to be \(\nabla (\mu /T)\)  with \(\mu =\mu (T,\rho )\)  being the particles' chemical potential [2]}, [5]}. For dilute systems one can assume the chemical potential to be equal to its excess Gibbs free energy and an ideal gas contribution
\(\mu (T,\rho ) = \Delta G(T) + k_{\text{B}}T \ln \frac{\rho }{\rho _0}\) 
